Microstructured fiber optic oscillator and waveguide for fiber scanner

ABSTRACT

Described are optical fibers and scanning fiber displays comprising optical fibers. The disclosed optical fibers include a plurality of mass adjustment regions, such as gas-filled regions, positioned between a central waveguiding element and an outer periphery for reducing a mass of the optical fiber as compared to an optical fiber lacking the plurality of mass adjustment regions.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/851,330 filed Dec. 21, 2017, now U.S. Pat. No. 10,451,868 B2 issuedOct. 22, 2019, entitled “MICROSTRUCTURED FIBER OPTIC OSCILLATOR ANDWAVEGUIDE FOR FIBER SCANNER,” which is a non-provisional of and claimsthe benefit of and priority to U.S. Provisional Application No.62/438,898, filed on Dec. 23, 2016, and U.S. Provisional Application No.62/464,298, filed on Feb. 27, 2017. All of the these applications arehereby incorporated by reference in their entireties.

BACKGROUND OF THE INVENTION

Optical fibers have been employed for a variety of uses, includingcommunication, sensors, and imaging. Optical fibers of variousconstructions exist and generally include a waveguide structure, such asa waveguide made of a central core and a surrounding cladding layer,with additional buffer and jacket layers optionally included to provideprotection during handling or exposure to environmental conditions.Additional optical fiber designs and optimizations are needed to improveand expand the variety of applications that optical fibers are or can beemployed in.

SUMMARY OF THE INVENTION

This application relates to optical waveguides. More specifically, andwithout limitation, this application relates to optical fibers andoptical fiber oscillators, such as used for scanning fiber displays,where the optical fibers include a waveguiding element and a mechanicalregion with one or more mass reduction elements positioned between thewaveguiding element and an outer periphery of the optical fiber. Theinclusion of the mass reduction elements advantageously provide ascanning fiber display incorporating the optical fiber with animprovement to a field-of-view, such as when compared to use ofconventional fiber optic oscillators.

Scanning devices generally trade off scanning range for frequency. Forexample, in general, as frequency increases, scanning range decreases.Similarly, as scanning range increases, frequency decreases. In manyapplications, such as scanning optical projectors, it is desirable,however, to have large operating frequency and large range. Frequencymay be important for both resolution and refresh rate. For example, in ascanning fiber display, the frequency may directly impact the refreshrate, as the repeated oscillations of a fiber may dictate how frequentlythe output view can be changed.

Range, however, may be important for field-of-view for a given projectordesign. For example, the maximum amplitude or range of an oscillatingfiber may provide for a limit on how wide an output image generated bythe fiber may be. As the oscillation range is increased, a wider fieldof view may be provided.

Scanning devices may also be useful as display devices due to theirsmall form factor and useful resolution and field-of-view. However, inorder to obtain high frequency scanning devices with a high scanningrange, innovations in this field are required. The presently describedoptical fibers allow for improved field-of-view projectors whilemaintaining a small form factor. As an example, by incorporating thedisclosed optical fibers into a scanning fiber display projector, thefield-of-view of the projector may be increased relative to conventionalscanning fiber display devices.

In a first aspect, provided are optical fibers. The disclosed opticalfibers may also be referred to herein as microstructured optical fibers.Example optical fibers include those comprising a waveguiding elementextending along an axis; a mechanical region surrounding the waveguidingelement, such as a mechanical region that is positioned between thewaveguiding element and an outer periphery and that comprises a firstmaterial having a first density; and a plurality of mass adjustmentregions positioned within the mechanical region, such as a plurality ofmass adjustment regions that comprise a second material having a seconddensity less than the first density. Such mass adjustment regions mayoptionally comprise air or may correspond to regions where material isremoved or otherwise absent from the mechanical region. It will beappreciated that the first and second materials may also have differentoptical properties, such as different indices of refraction.

As another example, the disclosed optical fibers include optical fiberscomprising a waveguiding element extending along an axis; a mechanicalregion surrounding the waveguiding element, such as a mechanical regionthat is positioned between the waveguiding element and an outerperiphery and that comprises a first material; and a plurality of secondmoment of area adjustment regions positioned within the mechanicalregion, such as a plurality of second moment of area adjustment regionsthat serve to modify the overall second moment of area of the mechanicalregion as compared to an identical optical fiber except that acorresponding mechanical region of the identical optical fiber does notinclude second moment of area adjustment regions positioned between acorresponding waveguiding element and a corresponding outer periphery ofthe identical optical fiber. As an example, second moment of areaadjustment regions may exhibit different mass per unit cross-sectionalarea than that of the first material and result in modification of theoverall second moment of area of the mechanical region. As a furtherexample, second moment of area adjustment regions may exhibit differentdensities than that of the first material and result in modification ofthe overall second moment of area of the mechanical region. It will beappreciated that the term second moment of area refers to a geometricalproperty of an area or object, as is known in the field of mechanicalengineering, and that other terms may be used interchangeably for secondmoment of area, including area moment of inertial, second area moment,and moment of inertia of plane area.

A variety of waveguiding elements are useful with the optical fibersdescribed herein. A waveguiding element may comprise a central coreregion and a cladding layer surrounding the central core region.Optionally, the central core region has a diameter of about 5 μm toabout 25 μm. Optionally, the cladding layer has a diameter of about 5 μmto about 200 μm. Optionally, the cladding layer comprises the firstmaterial. Optionally, the central core region comprises a thirdmaterial. Optionally, the central core region comprises the secondmaterial. Optionally, the cladding layer comprises a third material.Optionally, the cladding layer and the mechanical region comprise aunitary body. For example, the mass adjustment regions may optionally bepositioned within or as part of the cladding layer.

Optionally, the waveguiding element corresponds to a single modewaveguiding element or a multimode waveguiding element. Other usefulwaveguiding elements include those comprising a plurality of coreregions and a cladding layer surrounding the plurality of core regions.Optionally, each of the plurality of core regions may be the same ordifferent materials. Other waveguiding elements are contemplated,including those comprising a hollow (i.e., evacuated) or gas- orair-filled region, such as a gas-filled core region. It will beappreciated that hollow or gas- or air-filled cores may be useful inhigh power applications as gas or air may absorb less energy than glassor another solid material. Optionally, evacuated regions (i.e., vacuumfiled regions) may also be utilized. It will also be appreciated thatcore and cladding regions may exhibit different optical properties, suchas different indices of refraction.

A variety of mass adjustment regions may be employed with the opticalfibers described herein. For example, the mass adjustment regions mayinclude, but are not limited to, one or more gas- or air-filled regions,one or more polymer-filled regions, one or more glass-filled regions,one or more evacuated regions, or any combination of these. As anexample, the mechanical region may comprise a first glass and the massadjustment regions may comprise a second glass that is different fromthe first glass. Optionally, the plurality of mass adjustment regionscomprises a plurality of rows of mass adjustment elements. For example,the plurality of rows may be arranged concentrically around the centralwaveguiding element. Optionally, the plurality of mass adjustmentregions are arranged in a symmetric configuration around the axis.Optionally, each of the plurality of mass adjustment regions has acircular cross-sectional shape, an oval cross-sectional shape, or apolygonal cross-sectional shape. Combinations of cross-sectional shapesmay also be utilized. Optionally, each mass adjustment region has across-sectional shape with a lateral dimension or a diameter of about 1μm to about 25 μm. Optionally, the plurality of mass adjustment regionstraverse a length of the optical fiber, such as where each massadjustment region has its own longitudinal axis. Optionally, eachlongitudinal axis is arranged with axes parallel to the axis of theoptical fiber. Other configurations are possible, including whereindividual cells or regions of mass reduction material are included insections of the optical fiber. The mass adjustment regions mayoptionally run the entire length of the optical fiber, or only a portionof the fiber. Alternatively, the mass adjustment regions are randomly orevenly distributed throughout the mechanical region, or runperpendicular to or angled from an optical or waveguide axis.Optionally, a pitch between the plurality of mass reduction regions isabout 1 μm to about 25 μm. Optionally, the plurality of mass adjustmentregions occupy between about 30% and about 90% of a volume of themechanical region. Such a fractional or percentage volume may bereferred to herein as a mass reduction fraction or mass reduction filledfraction. In the case of a mass reduction region comprising air or gas,such a fractional or percentage volume may be referred to as anair-filled fraction or gas-filled fraction.

Optionally, the optical fiber comprises a composite optical fiber havinga plurality of different cross-sectional configurations. For example,the optical fiber may comprise a first segment comprising a firstcross-sectional configuration and a second segment comprising a secondcross-sectional configuration. In this way, an optical fiber maycomprise a segment that is microstructured and a segment that is notmicrostructured. Segmented optical fibers may be manufactured as asingle fiber with a varying cross-sectional configuration. Segmentedoptical fibers may also be constructed by splicing optical fibers ofdifferent cross-sectional configurations.

It will be appreciated that inclusion of mass-reduction regions mayallow for selection, tuning, or otherwise modifying the mechanicalproperties of an optical fiber. For example, an outside diameter of theoptical fiber may be proportional to a pointing angle of the opticalfiber. Optionally, a mass adjustment filled fraction of the mechanicalregion is proportional to a pointing angle of the optical fiber.Optionally, the mass adjustment filled fraction is represented by aratio of a diameter of the mass adjustment regions to a pitch betweenthe mass adjustment regions.

It will be appreciated that the plurality of mass adjustment regions mayreduce a mass of the optical fiber per unit length as compared to acomparable optical fiber comprising a corresponding waveguiding elementthat is identical to the waveguiding element and a correspondingmechanical region that is identical to the mechanical region except thatthe corresponding mechanical region does not include mass adjustmentregions positioned between the corresponding waveguiding element and acorresponding outer periphery of the comparable optical fiber.

Optical fibers may exhibit an effective cantilever length. Optionally,the plurality of mass adjustment regions increases a resonantoscillatory frequency of the optical fiber as compared to a comparableoptical fiber having the effective cantilever length and comprising acorresponding waveguiding element that is identical to the waveguidingelement and a corresponding mechanical region that is identical to themechanical region except that the corresponding mechanical region doesnot include mass adjustment regions positioned between the correspondingwaveguiding element and a corresponding outer periphery of thecomparable optical fiber. Optionally, the plurality of mass adjustmentregions increases an effective cantilever length of the optical fiberfor a given operating or resonant frequency as compared to a comparableoptical fiber comprising a corresponding waveguiding element that isidentical to the waveguiding element and a corresponding mechanicalregion that is identical to the mechanical region except that thecorresponding mechanical region does not include mass adjustment regionspositioned between the corresponding waveguiding element and acorresponding outer periphery of the comparable optical fiber.

Optical fibers, such as those having an effective cantilever length, mayhave a resonant frequency. Optionally, the plurality of mass adjustmentregions increases an effective cantilever length of the optical fiber ascompared to a comparable optical fiber having the resonant frequency andcomprising a corresponding waveguiding element that is identical to thewaveguiding element and a corresponding mechanical region that isidentical to the mechanical region except that the correspondingmechanical region does not include mass adjustment regions positionedbetween the corresponding waveguiding element and a corresponding outerperiphery of the comparable optical fiber.

In another aspect, scanning fiber displays are provided. For example, ascanning fiber display may optionally comprise any of the optical fibersdescribed above and an actuator in mechanical contact with the opticalfiber, the actuator for inducing an oscillation of the optical fiber. Asan example, an optical fiber in a scanning fiber display may optionallycomprise a waveguiding element extending along an axis; a mechanicalregion surrounding the waveguiding element, such as a mechanical regionthat is positioned between the waveguiding element and an outerperiphery and that comprises a first material having a first density;and a plurality of mass adjustment regions positioned within themechanical region, such as a plurality of mass adjustment regions thatcomprise a second material having a second density less than the seconddensity.

Various actuators and actuator configurations are useful with thescanning fiber displays described herein. For example, the actuatoroptionally comprises a piezoelectric transducer, an electromagneticvoice coil, or a thermal actuator. Optionally, the actuator comprises atwo-dimensional actuator for controlling motion of an end of the opticalfiber in two dimensions. Useful actuators include those that oscillateat a controllable frequency and may be configured to operate at or abouta natural or resonant frequency of an optical fiber.

The disclosed scanning fiber displays may optionally further comprise avisible optical source in optical communication with the waveguidingelement of the optical fiber. For example, a multi-color switchableoptical source in optical communication with the waveguiding element ofthe optical fiber may be used. In this way, color images may be outputby the scanning fiber display by controlling the light input into thewaveguiding element, such as by adjusting a color, or intensity, as afunction of the position of the optical fiber.

The foregoing, together with other features and embodiments, will becomemore apparent upon referring to the following description, claims andaccompanying drawings. It will be appreciated that the optical fibersand scanning fiber displays of the above aspects may optionally includefeatures and aspects described in the below description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A and FIG. 1B provide schematic illustrations of example opticalfiber systems in accordance with some embodiments.

FIG. 2A provides a schematic illustration of a cross-section of anexample conventional optical fiber. FIG. 2B provides a schematicillustration of a cross-section of an example microstructured opticalfiber.

FIG. 3A, FIG. 3B, FIG. 3C, and FIG. 3D provide schematic illustrationsof different cross sections for a microstructured optical fiber.

FIG. 4A and FIG. 4B provide schematic illustrations of example opticalfiber systems, showing a comparison between using a conventional opticalfiber and a microstructured optical fiber.

FIG. 5A provides a schematic illustration of a spiral output patternachieved by a scanning fiber display using a conventional optical fiber.FIG. 5B provides a schematic illustration of a spiral output patternachieved by a scanning fiber display using a microstructured opticalfiber.

FIG. 6 provides a plot showing gain in pointing angle of an opticalfiber as a function of the ratio of diameter to pitch of mass reductionregions and as a function of the diameter of mass reduction regions.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

Described herein are embodiments of optical fibers, fiber opticoscillators, and scanning fiber displays. The disclosed optical fibersadvantageously provide an improvement to the oscillation amplitude orpointing angle for a fixed oscillation frequency or resonant frequency,such as when compared to fiber optic oscillators having the same fixedoscillation frequency or resonant frequency but making use of aconventional optical fiber.

The disclosed optical fibers possess mechanical characteristicsdifferent from those of conventional fibers due to their constructionand materials properties. For example, a conventional optical fiber mayinclude a core region and a cladding region to define a waveguidingelement. These regions may be solid bodies of optical materialspossessing different indices of refraction so as to achieve totalinternal reflection and waveguiding of an optical beam down an axis ofthe optical fiber.

The optical fibers disclosed herein, also referred to as microstructuredoptical fibers, may optionally make use of a similar waveguiding elementof materials having different refractive indices, for example, but theyalso include a mechanical region surrounding the waveguiding element,such as a mechanical region that is not primarily used for waveguidingan optical beam but instead is used to tune, select, or otherwise modifythe mechanical properties of the optical fiber, such as to achievedesired mechanical properties. As an example, one or more second momentof area adjustment regions may be included in the mechanical region,which may serve to modify the second moment of area of the optical fiberas compared to an optical fiber that is identical except that it doesnot include the one or more second moment of area adjustment regions. Ina specific example, the second moment of area may be adjusted bymodifying the mass of the mechanical region. For example, one or moremass adjustment regions may be included in the mechanical region, whichmay serve to reduce the mass or mass per unit length of the opticalfiber as compared to an optical fiber that is identical except that itdoes not include the one or more mass reduction regions. Example massadjustment regions include air-filled regions (or other gas-filledregions) or regions comprising other materials that have a density lessthan that of a material used for the waveguiding element or themechanical region. For example, plastics, polymers, or glasses havingdensities less than the material used in the waveguiding element or themechanical region may be employed. This reduction in mass, may, forexample, allow for optical fibers of desired mechanical properties to becreated and used. In addition, the reduction in mass may correspond tomodification of the moment of area of the mechanical region.

It will be appreciated that identical optical fibers may refer to twooptical fibers having identical geometries, materials, and/orconstructions, and reference to an exception between identical opticalfibers may indicate that the exception is one characteristic of onefiber that is different from the other fiber, such as one optical fiberbeing microstructured and one optical fiber not being microstructured.For example, an optical fiber may include a core, such as a core havinga first cross-sectional dimension (such as a diameter) and made of afirst optical material, and a cladding surrounding the core, such as acladding having a second cross-sectional dimension (such as an outerdiameter) and made of a second optical material. An optical fiber thatis identical except that it includes one or more mass adjustmentregions, such as air- or-gas filled regions, may refer to amicrostructured optical fiber that includes a core, such as a corehaving the first cross-sectional dimension and made of the first opticalmaterial, a cladding surrounding the core, such as a cladding having thesecond cross-sectional dimension and made of the second opticalmaterial, and one or more mass adjustment regions located in thecladding. It will be appreciated that identical optical fibers may haveother characteristic differences aside from the mass reduction regionsthat arise due to the presence of the mass reduction regions, such asdifferent mass per unit length or different resonant frequency for afixed oscillating fiber length or different oscillating fiber length fora fixed resonant frequency.

It will further be appreciated that identical optical fibers may haveslightly different characteristics depending on whether certainattributes are the same between identical optical fibers. For example,two optical fibers that are identical, except for the inclusion of amicrostructured mechanical region, and that have a same oscillatinglength will have different resonant frequencies, such as where themicrostructured optical fiber has a higher resonant frequency. Asanother example, two optical fibers that are identical, except for theinclusion of a microstructured mechanical region, and that have the sameresonant frequencies will have different oscillation length, such aswhere the microstructured optical fiber has a longer oscillation length.

Advantageously, the disclosed optical fibers may provide for an improvedfield of view of a scanning fiber display for a given scanningfrequency. For example, a scanning fiber display that uses amicrostructured optical fiber including a mechanical region includingone or more mass reduction regions may have an increased field of viewas compared to the field of view of a scanning fiber display that usesan identical optical fiber with a same resonant frequency except that itdoes not include one or more mass reduction regions (i.e., anon-microstructured optical fiber). Since the field of view may be alimiting factor in consumer acceptance of augmented reality devices,increasing the field of view may be beneficial for increasing consumeradoption. It will be appreciated that, in some scanning fiber displayembodiments, the field of view may be increased by increasing the lengthof the oscillating fiber for a given operating frequency, as this willresult in an increase in the maximum pointing angle of the oscillatingfiber.

FIG. 1A provides a schematic illustration of an example optical fibersystem 100. Example optical fiber system includes an optical source 105,coupling optics 110, and optical fiber 115. Optical source 105 mayinclude a light emitting diode, a laser, or other visible opticalsource, for example. Optical source 105 may optionally include aplurality of sub-sources or a multi-color optical source, such assources outputting different wavelengths of electromagnetic radiation.In embodiments, optical source 105 may be switchable, such as to allowfor control over the output or intensity of the optical source 105 as afunction of time.

Coupling optics 110 may include one or more optical elements, such aslenses, mirrors, reflectors, etc., arranged in a configuration to enablelight from optical source 105 to be suitably directed into the core 120of optical fiber 115 for waveguiding. Thus, optical source 105 may bepositioned in optical communication with a waveguiding element ofoptical fiber 115. It will be appreciated that the coupling opticsneeded to efficiently couple light from optical source 105 may bedependent upon optical source 105 and the geometry, materials, and/orthe numerical aperture of the optical fiber 115.

As illustrated, optical fiber 115 includes core 120 and cladding 125 andhas an axis 130, which may correspond to an optical axis or awaveguiding axis, for example. Light from optical source 105 that iscoupled into core 120 and waveguided along the length of optical fiber115 may be output at the opposite end of optical fiber 115. It will beappreciated that the spot shape and direction of the light output fromoptical fiber 115 may be dependent upon the geometry, materials, and/orthe numerical aperture of the optical fiber 115. Typically, output froman optical fiber exhibits a cone shape 135, with the angle of the cone135 again defined by the geometry, materials, and/or the numericalaperture of the optical fiber 115. In terms of field-of-view, opticalfiber 115, in a non-oscillatory configuration, exhibits no increase infield-of-view 140 beyond the angle of cone 135. In terms of deflectionangle, optical fiber 115, in a non-oscillatory configuration, exhibits adeflection angle of zero.

FIG. 1B provides a schematic illustration of an optical fiber system150, such as may be present in a scanning fiber display system. Generaldetails of a scanning fiber display system may be found, for example, inU.S. patent application Ser. No. 14/156,366, filed on Jan. 15, 2014 andpublished under publication no. US 2015/0268415, which is herebyincorporated by reference in its entirety.

FIG. 1B omits depiction of any optical source or coupling elements fromoptical fiber system 150 so as not to obstruct other details. Opticalfiber system 150 includes optical fiber 155, which may correspond tooptical fiber 115, and actuator 160. Actuator 160 may be used to impartoscillatory motion into optical fiber 155. Oscillations of optical fiber155 may be modeled as or correspond to a cantilevered oscillator with afixed end and a free end. Actuator 160 may be or include a piezoelectricactuator, an electromagnetic voice coil, or a thermal actuator, forexample. Actuator 160 may allow for control over oscillatory motion ofoptical fiber 155 in two dimensions and may include two or moreindependent actuatable axes. The extent of the oscillatory motion ofoptical fiber 155 is depicted with dashed lines in FIG. 1B. Due to theoscillatory motion, optical fiber 155 exhibits an increase in field ofview beyond the output cone of optical fiber 155. In terms offield-of-view, optical fiber system 150 exhibits a field-of-view 165that is greater than the output cone of optical fiber 155.

FIG. 2A provides a schematic cross-sectional illustration of an opticalfiber 200. Optical fiber 200 may correspond to a conventional opticalfiber, and includes a core 205 and a cladding 210 surrounding core 205.Core 205 is illustrated as having a core diameter 215 and cladding 210is illustrated as having an outer diameter 220. It will be appreciatedthat the core diameter 215 and outer diameter 220 may be characteristicof a particular optical fiber embodiment, and thus may take on anysuitable values.

It will be appreciated that, unless otherwise indicated, the dimensionsof features illustrated in the accompanying drawings may not be toscale, though certain aspects of a figure or different figures aredepicted to illustrate a difference in a dimension between differentconfigurations or elements. It will also be appreciated that additionalmaterials, such as a buffer, jacket, or other coated or protectivematerials may be constructed outside of the cladding or mechanicalregion, but may not be shown in the accompanying figures.

FIG. 2B provides a schematic cross-sectional illustration of anembodiment of a microstructured optical fiber 230. Microstructuredoptical fiber 230 includes a waveguiding element 235 and a mechanicalregion 240 surrounding the waveguiding element. For illustrationpurposes, a dashed line is shown in FIG. 2B to better identify thetransition between waveguiding element 235 and mechanical region 240. InFIG. 2B, waveguiding element 235 includes a core 245 and a cladding 250surrounding core 245. Core 245 is depicted as having a core diameter255, waveguiding element 235 is depicted as having a cladding diameter260, and optical fiber 230 is depicted as having an outer diameter 265.

Mechanical region 240 is depicted in FIG. 2B as including solid regions270 and mass reduction regions 275 positioned between waveguidingelement 235 and an outer periphery of mechanical region 240 andmicrostructured optical fiber 230. Example mass adjustment regionsinclude, but are not limited to fluid-filled regions, gas- or air-filledregions, polymer-filled regions, glass-filled regions, and/or evacuatedregions (e.g., vacuum-filled), with the fluid-, gas- or air-, polymer-,glass-filled, or evacuated regions having a density less than that ofsolid regions 270, cladding 250, core 245, or any combination of these.Optionally, solid regions 270 comprise the same material as cladding 250and/or have similar or identical optical and/or mechanical properties.Optionally, solid regions 270 and cladding 250 comprise differentmaterials and or have different optical and/or mechanical properties. Itwill be appreciated that additional materials, such as a buffer, jacket,or other coated or protective materials may be constructed outside ofthe outer periphery of mechanical region 240, but are not illustratedhere.

Mass reduction regions 275 may be uniformly and/or regularly distributedthroughout mechanical region 240 and any suitable or desirablegeometries and distribution may be used in order to obtain particularmechanical properties of interest for microstructured optical fiber 230.It will be appreciated that mass reduction regions 275 may be arrangedalong axes parallel to one another and/or parallel to an axis of theoptical fiber, such as a waveguiding axis or an optical axis.Optionally, mass reduction regions 275 may be arranged along otherdirections, such as along intersecting axes, perpendicular to theoptical axis, or angled with respect to the optical axis, however atleast a portion of the mechanical region 240 includes mass reductionregions. The mass reduction regions 275 may also be randomly, evenly, orunevenly distributed (optionally along no particular axis) throughoutthe mechanical region 240. As illustrated in FIG. 2B, mass reductionregions 275 exhibit uniform cross-sections, which are shown as circularand having a diameter 280. Pitch 285 corresponds to the center-to-centerspacing between adjacent mass reduction regions 275. Mass reductionregions 275 may exhibit a symmetry, such as a cylindrical symmetry,about an axis of microstructured optical fiber 230, such as awaveguiding axis or optical axis. The optical fiber 230 may optionallyexhibit rotational symmetry.

Without limitation, microstructured optical fiber 230 may be constructedby stacking lengths of materials of appropriate sizes to form an overallpreform structure targeted for generating microstructured optical fiber230, such as by using solid tubes and/or hollow tubes of suitablediameters, wall thickness, materials, shape, etc. In some embodiments,glass materials are used. Example glasses may include, but are notlimited to silica glasses, fluoride glasses, phosphate glasses,chalcogenide glasses. In some embodiments, plastics or polymers may beused, such as polymethyl methacrylate, polystyrene, fluoropolymers, orpolysiloxanes. Depending on the fabrication method and materials, thepreform may be placed in a furnace to heat and fuse the differentcomponents of the preform and the heated preform may be drawn into astrand of optical fiber. Optionally, extrusion methods may be used, suchas for fibers comprising polymer or plastic materials. It will beappreciated that various techniques, materials, and methods may be usedto manufacture an optical fiber and a number of commercial fibermanufacturers exist and may provide services for manufacturing opticalfibers based on specified parameters.

For illustrative purposes of comparison, core diameter 215 of core 205of optical fiber 200 may optionally be the same as core diameter 255 ofcore 245 of microstructured optical fiber 230. Outer diameter 220 ofoptical fiber 200 may optionally be the same as outer diameter 265 ofmicrostructured optical fiber 230. Core 205 and core 255 may optionallybe composed of the same material. Cladding 210 and cladding 250 andsolid regions 270 (non-mass reduction regions) of mechanical region 240may optionally be composed of the same material. In this respect,optical fiber 200 may be considered identical to microstructured opticalfiber 230 except that microstructured optical fiber 230 includes massreduction regions 275, while optical fiber 200 includes a solid cladding210 that does not include mass reduction regions.

The different components of microstructured optical fiber 230 may takeon any suitable dimensions and certain dimensions may be selected toprovide particular properties, such as optical properties and mechanicalproperties. For example, core 245 may have, but is not limited to, adiameter of about 5 μm to about 25 μm. It will be appreciated that theterm about, as used herein, is intended to include a variation around aspecified value, such as a variation that would not modify theoperational effect if the value were slightly smaller or slightlylarger. In some embodiments, the term about may relate to a precision ortolerance of a value. In some embodiments, the term about may correspondto a variation of ±1% or less, a variation of ±5% or less, or avariation of ±10% or less.

As another example, the waveguiding element 235 may have, but is notlimited to, a diameter of about 5 μm to about 200 μm, such as about 5 μmto about 125 μm. In some embodiments, cladding 250 may have, but is notlimited to, a diameter or thickness of about 5 μm to about 200 μm, suchas about 5 μm to about 125 μm, and may optionally be considered toencompass or may be integral or a unitary body with mechanical region240 and thus may have a diameter or thickness corresponding to outerdiameter 265. Outer diameter 265 may also take on any suitable value,such as about 10 μm to about 200 μm, and may match the outer diameter220 of conventional optical fiber 200. Example outer diameters mayinclude about 40 μm, about 50 μm, about 80 μm, about 100 μm, about 125μm, about 150 μm, about 175 μm, and about 200 μm.

Each of mass reduction regions 275 may take on any suitable dimensionsor shapes, and may, for example, have, but is not limited to, across-sectional dimension, such as diameter, radius, side length, oraxis length, of about 1 μm to about 25 μm, about 1 μm to about 5 μm,about 5 μm to about 10 μm, about 10 μm to about 15 μm, about 15 μm toabout 20 μm, or about 20 μm to about 25 μm. Pitch 280 between massreduction regions 275 may also take on any suitable dimensions, and maybe limited by the cross-sectional dimensions of mass reduction regions275. For example, pitch 280 may be greater than a diameter of massreduction regions 275. Pitch 280 may have, but is not limited to, alength of between about 1 μm to about 25 μm, about 1 μm to about 5 μm,about 5 μm to about 10 μm, about 10 μm to about 15 μm, about 15 μm toabout 20 μm, or about 20 μm to about 25 μm. The mass reduction fractionof optical fiber 230 and/or mechanical region 240 may take on anysuitable value based on the size, number, spacing, and arrangement ofmass reduction regions. In embodiments, the plurality of mass reductionregions occupy between about 1% and 90% of the volume of optical fiber230 or of the volume of mechanical region 240. Optionally, the pluralityof mass reduction regions occupy between about 30% and about 90%, about30% or greater, about 40% or greater, about 50% or greater, about 60% orgreater, about 70% or greater, about 80% or greater or about 90% of thevolume of optical fiber 230 or of the volume of mechanical region 240.

Depending on the particular configuration, in some embodiments, massreduction regions may exhibit a 4-fold or 6-fold or other symmetry, suchas cylindrical symmetry, rotational symmetry, or radial symmetry, aboutan axis of a microstructured optical fiber. In addition, othercross-sectional shapes for mass reduction regions may be utilized. Forexample, the cross-section of a mass reduction region may exhibit apolygonal shape, such as a triangle, square, rectangle, hexagon, etc., around, circular, or oval shape, or any other suitable shape. In someembodiments, the cross-section of a mass reduction region may have shapewith a regular symmetry, such as a circle, oval, ellipse, polygon, etc.In embodiments, combinations of different cross-sectional shaped massreduction regions may be utilized. In embodiments, a spacing betweenadjacent mass reduction regions may be uniform or non-uniform. Inembodiments, the cross-sectional dimensions, such as a diameter, radius,axis length, side length, etc., of different mass reduction regions maybe uniform or non-uniform.

FIGS. 3A-3D depict schematic cross-sectional illustrations of differentmicrostructured optical fibers that exhibit a waveguiding elementsurrounded by a mechanical region in accordance with variousembodiments. Microstructured optical fiber 300A in FIG. 3A includes aplurality of rows of mass reduction regions arranged concentricallyaround the central waveguiding element. Microstructured optical fiber300B in FIG. 3B includes circular mass reduction regions arranged in a6-fold symmetric configuration. Microstructured optical fiber 300C inFIG. 3C includes square mass reduction regions arranged in a 4-foldsymmetric configuration and a plurality of core regions. Microstructuredoptical fiber 300D in FIG. 3D includes concentric rings of oval-shapedmass reduction regions.

Although the waveguiding elements of the microstructured optical fibersdescribed above correspond to conventional core/cladding designs, otherwaveguiding element configurations are possible and contemplated. Forexample, in some embodiments, multiple core regions may be surrounded bya single cladding layer or region. In addition, multiple optical fibersmay be arranged in side-by-side or in a two-dimensional arrayconfiguration to provide additive fields-of-view for a scanning fiberdisplay (also referred to as a fiber scanned display). U.S. patentapplication Ser. No. 14/156,366 describes hexagonally packed multicorefibers, such as including 7 or 19 cores in a hexagonally arrangedconfiguration, as well as arrays of oscillating fibers for a fiberscanned display. It will be appreciated that, in embodiments, a changeof refractive index between materials, such as between core and claddingor between glass and air, can provide for a waveguiding effect, such asby the process of total internal reflection. As such, a transitionbetween materials may be all that is needed to achieve waveguiding, andthus the large cladding of a conventional optical fiber may be modifiedto include mass reduction regions while still retaining a solid centralportion surrounding a core of a higher refractive index material thatprovides for waveguiding. Various fiber configurations are possible,including single-mode configuration and multi-mode configuration. Forpurposes of generating an optical display, it is beneficial for theoptical fiber to have a high transparency/low loss in the visibleelectromagnetic region.

FIG. 4A and FIG. 4B provide schematic illustrations of optical fibersystems 400 and 450 for comparative purposes. Optical fiber systems 400and 450 may be used, for example, in a scanning fiber display system.FIGS. 4A and 4B omit depiction of any optical source or couplingelements from optical fiber systems 400 and 450 so as not to obstructother details. In FIG. 4A, optical fiber system 400 includes opticalfiber 405 and actuator 410. The cross-section 415 of optical fiber 405is also shown in FIG. 4A, illustrating that optical fiber 405 is aconventional optical fiber including a core and a cladding layer.Actuator 410 may be used to impart oscillatory motion into optical fiber405. The extent of the oscillatory motion of optical fiber 405 isdepicted with dashed lines in FIG. 4A. In terms of field-of-view,optical fiber system 400 exhibits a field-of-view 420. The cantileveredportion of optical fiber 405 has a length 425.

In FIG. 4B, optical fiber system 450 includes microstructured opticalfiber 455 and actuator 460. The cross-section 465 of microstructuredoptical fiber 455 is also shown in FIG. 4A, illustrating thatmicrostructured optical fiber 465 includes a waveguiding element and amicrostructured mechanical region, surrounding the waveguiding element,that includes a plurality of mass reduction regions. Actuator 460 may beused to impart oscillatory motion into microstructured optical fiber455. Actuator 460 may oscillate at our about at a resonant frequency ofmicrostructured optical fiber 455, such as within 5% of the naturalfrequency of microstructured optical fiber 455, or within 1% of aresonant frequency of microstructured optical fiber 455. In embodiments,a resonant frequency may correspond to an eigenfrequency or a naturalfrequency. In embodiments, actuator 460 operates at a frequency thatprovides a gain in pointing angle or deflection of the microstructuredoptical fiber 455. The extent of the oscillatory motion ofmicrostructured optical fiber 455 is depicted with dashed lines in FIG.4B. In terms of field-of-view, microstructured optical fiber system 450exhibits a field-of-view 470. The cantilevered portion ofmicrostructured optical fiber 455 has a length 475.

It will be appreciated that the resonant frequency of a cantileveredoptical fiber may generally be proportional to the square of the lengthof the cantilever. For example, if the length 425 of optical fiber 405is doubled, an increase in the resonant frequency of optical fiber 405by a factor of 4 would be expected. Similarly, if the length 475 ofmicrostructured optical fiber 455 is halved, a decrease in the resonantfrequency of microstructured optical fiber 475 by a factor of 4 would beexpected.

It will also be appreciated that the distribution of the mass of thecantilevered optical fiber may also impact the resonant frequency. Forexample, assuming that optical fiber 405 and microstructured opticalfiber 455 are identical (diameters, materials, etc.), except for themass reduction regions of microstructured optical fiber 455, theinclusion of the mass reduction regions may reduce the mass per unitlength of microstructured optical fiber 455 as compared to optical fiber405. Thus, in order for the resonant frequencies of optical fiber 405and microstructured optical fiber 455 to be the same, optical fibers 405and microstructured optical fiber 455 will exhibit different lengths,with length 425 being smaller than length 475. Advantageously, thisdifference in length will allow field-of-view 470 and/or pointing angleof microstructured optical fiber 455 to be greater than the field ofview 420 and/or pointing angle of optical fiber 405.

It will be appreciated that other characteristics of an optical fibermay impact the resonant frequency and/or pointing angle, such as maximumpointing angle, of the optical fiber. Example characteristics that mayimpact a frequency or pointing angle include the fiber outer diameter,the diameter of the waveguiding element, the mass reduction fraction,the quantity, distribution, and cross-sectional dimensions (e.g.,diameters) of the mass reduction regions, the pitch between adjacentmass reduction regions, mass reduction region material densities, thewaveguiding element design, the core material, the cladding material,the solid material of the mechanical region, and the like.

In some embodiments, a scanning fiber display makes use of theoscillatory motion of a cantilevered optical fiber to project an imageusing an optical fiber. For example, the oscillatory motion of acantilevered optical fiber may be controlled in two dimensions togenerate a spiral pattern, such as by appropriately driving an actuator.In some embodiments, the input light may be controlled and timed so thatthe output of the oscillating optical fiber may generate a desired imagewithin the spiral pattern, with repeated oscillatory motion and timedoptical outputs used to generate a sequence of images. U.S. patentapplication Ser. No. 14/156,366 describes how a cantilevered fiber maybe used to generate a projected image or sequence of images. Theembodiments described herein, however, advantageously allow thefield-of-view, pointing angle, and projected output image size and/orresolution of a scanning fiber display to be increased by using amicrostructured optical fiber.

For example, FIG. 5A depicts a spiral pattern 500 for a conventionalscanning fiber display incorporating a conventional optical fiber with acore and cladding and no mass reduction regions in the claddingmaterial, such as similar to optical fiber system 400 of FIG. 4A. Thediameter 505 of spiral pattern is limited by the maximum pointing angleof the scanning fiber display used.

In contrast, FIG. 5B depicts a comparable spiral pattern 550 for ascanning fiber display incorporating a comparable microstructuredoptical fiber including a waveguiding element and a microstructuredmechanical region, surrounding the waveguiding element, that includes aplurality of mass reduction regions, such as similar to optical fibersystem 450 of FIG. 4B. The spiral pattern 550 exhibits a diameter 555that is limited by the maximum pointing angle of the scanning fiberdisplay used.

For identical conventional and microstructured optical fibers (i.e.,identical except for inclusion of mass reduction regions in themicrostructured optical fiber) operating at the same resonant frequency(which implies increased cantilever length), the diameter 555 for spiralpattern 550 will be larger than the diameter 505 for spiral pattern 500,corresponding to the increase in field-of-view and/or maximum pointingangle achieved by use of a microstructured optical fiber. Withoutlimitation, use of a microstructured optical fiber may advantageouslyincrease a field-of-view and/or pointing angle by up to about 30%. Insome cases, an increase in field-of-view and/or pointing angle of up toabout 50%, or up to about 70% may be achieved. In some embodiments, anincrease in field-of-view and/or pointing angle of between about 30%about and 40% may be achieved.

Various microstructured optical fibers may have different optical andmechanical properties. In some embodiments, a microstructured opticalfiber may have one or more of the following optical specifications: anoptical transmission range of about 435 nm to about 645 nm; an outputmode field diameter for red, green, and/or blue light of about 1.4 μm,with a tolerance of about ±0.15 μm; a numerical aperture for red, green,and/or blue light of about 0.25; an optical transmission loss of lessthan or about 30 dB/km for any or all wavelengths between about 435 and645 nm; and a low splicing loss to single mode (Φ1.2 μm, NA).

In some embodiments, a microstructured optical fiber may have one ormore of the following mechanical specifications: an outer diameter ofbetween about 80 μm and about 125 μm or between about 40 μm and about200 μm; a diameter of the mechanical region of between about 40 μm andthe outside diameter; a mass reduction fraction (e.g., gas- orair-filled fraction) in the mechanical region of about 70% or greater; aconcentricity core/outer diameter of about 500 nm or less; a percentdifference between perpendicular moments of inertia of about 0.4% orless, indicating that the microstructured optical fiber is approximatelysymmetric in the x and y axes (where z is the fiber length axis); and aweight change due to water collecting in the air-filled regions of about1% or less.

It will be appreciated that rotational symmetry may be useful for theoptical fibers disclosed herein and may be advantageous for someembodiments and thus a microstructured optical fiber may optionallypossess rotational symmetry. A rotational symmetry may refer to thestiffness of the optical fiber (K_(r)) being the same in all radialdirections (θ) (see FIG. 2A for directional reference). Stated anotherway, for a force acting along any radial direction, defined by θ, theoptical fiber will translate purely in the radial direction. In thestatic case, Hooke's law gives F_(r)=K_(r)·δ_(r). With rotationalsymmetry, it is noted that there is no translation orthogonal to theradial direction. This can also be characterized by stating that theprincipal directions of the optical fiber are not unique.

The terms and expressions which have been employed are used as terms ofdescription and not of limitation, and there is no intention in the useof such terms and expressions of excluding any equivalents of thefeatures shown and described or portions thereof, but it is recognizedthat various modifications are possible within the scope of theinvention claimed. Thus, it should be understood that although thepresent invention has been specifically disclosed by preferredembodiments and optional features, modification and variation of theconcepts herein disclosed may be resorted to by those skilled in theart, and that such modifications and variations are considered to bewithin the scope of this invention as defined by the appended claims.

The above description of exemplary embodiments of the invention has beenpresented for the purposes of illustration and description. It is notintended to be exhaustive or to limit the invention to the precise formdescribed, and many modifications and variations are possible in lightof the teaching above. The embodiments were chosen and described inorder to explain the principles of the invention and its practicalapplications to thereby enable others skilled in the art to utilize theinvention in various embodiments and with various modifications as aresuited to the particular use contemplated.

When a group of substituents is disclosed herein, it is understood thatall individual members of those groups and all subgroups and classesthat can be formed using the substituents are disclosed separately. Whena Markush group or other grouping is used herein, all individual membersof the group and all combinations and subcombinations possible of thegroup are intended to be individually included in the disclosure. Asused herein, “and/or” means that one, all, or any combination of itemsin a list separated by “and/or” are included in the list; for example“1, 2 and/or 3” is equivalent to “‘1’ or ‘2’ or ‘3’ or ‘1 and 2’ or ‘1and 3’ or ‘2 and 3’ or ‘1, 2 and 3’”.

Every formulation or combination of components described or exemplifiedcan be used to practice the invention, unless otherwise stated. Specificnames of materials are intended to be exemplary, as it is known that oneof ordinary skill in the art can name the same material differently. Oneof ordinary skill in the art will appreciate that methods, deviceelements, starting materials, and synthetic methods other than thosespecifically exemplified can be employed in the practice of theinvention without resort to undue experimentation. All art-knownfunctional equivalents, of any such methods, device elements, startingmaterials, and synthetic methods are intended to be included in thisinvention. Whenever a range is given in the specification, for example,a temperature range, a time range, or a composition range, allintermediate ranges and subranges, as well as all individual valuesincluded in the ranges given are intended to be included in thedisclosure. The specific details of particular embodiments may becombined in any suitable manner without departing from the spirit andscope of embodiments of the invention. However, other embodiments of theinvention may be directed to specific embodiments relating to eachindividual aspect, or specific combinations of these individual aspects.

The invention may be further understood by reference to the followingnon-limiting examples.

Example 1: Description of Mechanical Merit Function: Cantilevered FiberOptic Oscillator

This example describes how to maximize pointing angle, and thus increasefield-of-view, at the end of the fiber optic while keeping the naturalfrequency of the oscillator constant. In some embodiments, this can beaccomplished by minimizing the mass of a cantilevered oscillator whilemaximizing the second moment of area. This example derives: for a givensize fiber, how the optimal performance (maximum deflection for a givenoperating frequency) is achieved for a thin walled tube fiber.Advantageously, microstructured optical fibers incorporating thistechnology can exhibit a merit function increase (i.e., increase inpointing angle), when compared to an identical traditional fiber, ofapproximately 30% or better.

To determine the merit function, the second moment of area of a sectionof optical fiber is needed. The following definitions are used in thecalculation of the second moment of area:

I_(T,MS): Second moment of area of cross section (T=standard fiber,MS=Microstructured Fiber)

A_(T,MS): Area of cross-section

A_(H_i): Area of hole (air-filled region) in microstructure

r_(i): Perpendicular distance from hole to neutral axis of area section

T: Subscript to indicate solid fiber

MS: Subscript to indicate microstructured fiber

j: Natural frequency mode/harmonic number

β(j): Mode constant for j^(th) mode

d: Rod diameter.

For a solid rod, I_(T)=d⁴/64. As a constraint, the natural frequency,f_(n), is fixed to be the same for direct comparison of the solid andmicrostructured designs. For a cantilevered beam the natural frequencyis obtained from the Euler equations:

$f_{n} = {\frac{1}{2\;\pi}{\beta \cdot L \cdot j^{2}}\sqrt{\frac{EI}{\rho\;{AL}^{4}}}}$

For high refresh rates of displays, it is useful that the operatingfrequency of the scanning fiber be high. At the same time, it isdesirable to achieve a large deflection of the oscillator as this isproportional to the field of view and resolution.

Since the scanning fiber projector is a resonant device, the operatingfrequency is about equal to the natural frequency (within 1%). Thismeans that we must choose the system parameters such that the systemnatural frequency, as estimated by the above equation, remains high.

From the perspective of the micro-structured fiber, the above equationgives direct insight to the effect of I, the second moment of area, andA, the solid cross-section area of solid fiber minus area of holes, onthe natural frequency and the associated length of the oscillator.

By inspection of this equation, increasing the ratio of I to A increasesthe natural frequency. This property can be exploited in microstructuredfibers by removing mass near the neutral axis (small effect on I) byinserting holes near the neutral axis, since this has a small effect onI due to being a function of the distance from the neutral axis squared,and keeping mass near the outside diameter or periphery. This furtherflows from the following equation, where it is shown that thecontribution of a section (hole) to the second moment of area is afunction of the distance from the neutral axis squared.

From the parallel axis theorem, the second moment of area for themicro-structured fiber is:

$I_{MS} = {I_{T} - {\sum\limits_{i = 1}^{N}{A_{H\_ i} \cdot r_{i}^{2}}}}$Thus, sections can be deleted (i.e., holes or reduced mass regionsinserted) furthest from the neutral axis to boost the natural frequency,or increase the oscillator length, L, for a given natural frequency.Inserting holes allows for boosting the oscillator natural frequency, orincreasing the oscillator length (L) for a given natural frequency. Notethat all holes have the same effect on the area (A) regardless of theirposition, while holes nearest the neutral axis boost the naturalfrequency more than holes that are farther away from the neutral axis

The below Merit Function analysis describes how a microstructured fibercompares to a solid fiber in terms of deflection angle. In thisanalysis, the frequency is held constant and the second moment of areaand area of the beam section is varied. The merit function quantifiesthe gain in the deflection angle. Using the above expression for thenatural frequency:

$f_{n_{i}} = {{\frac{1}{2\;\pi}{\beta \cdot L \cdot \left( n_{i} \right)^{2}}\sqrt{\frac{EI}{\rho\;{AL}^{4}}}} = {\frac{1}{2\;\pi}{\beta \cdot L \cdot \left( n_{i} \right)^{2} \cdot \frac{r_{g}}{L^{2}}}\sqrt{\frac{E}{\rho}}}}$Solving for the value of L that constrains the frequency to be aconstant yields:

${L^{4} = \frac{EI}{A\;{\rho\pi}^{2}\beta\;{L\left( n_{i} \right)}^{4}f_{n}^{2}}} = {\frac{I}{A} \cdot \frac{E}{{\rho\pi}^{2}\beta\;{L\left( n_{i} \right)}^{4}f_{n}^{2}}}$$L = {\sqrt[4]{\frac{I}{A}} \cdot \sqrt[4]{\frac{E}{{\rho\pi}^{2}\beta\;{L\left( n_{i} \right)}^{2}f_{n}^{2}}}}$Fixing the frequency (f_(n)), density (ρ), mode constant (βL(n_(i)))(comparing 1^(st) mode to 1^(st) mode, 2^(nd) mode to 2^(nd) mode,etc.), and modulus (E) constant, the above expression reduces asfollows:

$L = {{{{constant} \cdot \sqrt[4]{\frac{I}{A}}}\mspace{14mu}{or}\mspace{14mu} L} \propto \sqrt[4]{\frac{I}{A}}}$

From the frequency response function, the response at resonance isidentified as scaling with the static deflection. Thus, staticdeflection of beams under point and distributed loads are considered. Ineach case, the ratio of the deflection of a structured device to thedeflection angle of a solid fiber optic cantilever is determined. Note,the pattern of the microstructure can be optimized for max deflection,but it must also be tuned for transmission of visible light with asingle mode.

Computing the slope of a beam with concentrated load at end:

$\alpha = \frac{- {FL}^{2}}{2\;{EI}}$where α is the deflection angle, E is Young's Modulus, I is the secondmoment of area, L is the cantilever length, and F is force. Computingthe slope of a beam with a distributed load

$\alpha = \frac{- {WL}^{3}}{6{EI}}$where W is the load per unit length.

The expressions for the angular deflection gains for constant frequencyoscillators are derived as follows. For the concentrated force model,assuming E_(MS)=E_(T):

$\frac{\alpha_{MS}}{\alpha_{T}} = {{\frac{- {FL}_{MS}^{2}}{2\; E_{MS}I_{MS}} \cdot \frac{2\; E_{T}I_{T}}{- {FL}_{T}^{2}}} = \frac{L_{MS}^{2}I_{T}}{L_{T}^{2}I_{MS}}}$${L^{2} \propto \sqrt{\frac{I}{A}}} = {{r_{g}L_{MS}^{2}} \propto \sqrt{\frac{I_{MS}}{A_{MS}}}}$${\frac{\alpha_{MS}}{\alpha_{T}} \approx {\sqrt{\frac{I_{MS}A_{T}}{A_{MS}I_{T}}}\frac{I_{T}}{I_{MS}}}} = {{\frac{r_{g_{MS}}}{r_{g_{T}}}\;\frac{I_{T}}{I_{MS}}} = \sqrt{\frac{I_{T}A_{T}}{A_{MS}I_{MS}}}}$This represents the gain relationship, which allows for calculationusing the area and the second moment of area for the sections.

For the distributed force model:

$L^{3} \propto \left( \frac{I}{A} \right)^{\frac{3}{4}}$$L_{MS}^{3} \propto {I_{MS}/A_{MS}^{\frac{3}{4}}}$${\frac{\alpha_{MS}}{\alpha_{T}} \approx {\frac{L_{MS}^{3}}{L_{T}^{3}}\;\frac{I_{T}}{I_{MS}}}} = {{\left( \frac{I_{MS}A_{T}}{A_{MS}I_{T}} \right)^{\frac{3}{4}}\frac{I_{T}}{I_{MS}}} = {\left( \frac{A_{T}}{A_{MS}} \right)^{\frac{3}{4}}\left( \frac{I_{T}}{I_{MS}} \right)^{0.25}}}$This represents the gain relationship, which allows for calculationusing the area and the second moment of area for the sections.

Approximating the dynamic mode shape as that of a distributed load for arelative stress calculation:

$\frac{\sigma_{MS}}{\sigma_{T}} = \frac{z_{MS}\rho_{MS}A_{MS}L_{MS}^{2}I_{T}}{z_{T}\rho_{T}A_{T}L_{T}^{2}I_{T}I_{MS}}$${L_{MS}^{3} \propto \left( {I_{MS}/A_{MS}} \right)^{\frac{3}{4}}},{L_{T}^{3} \propto \left( {I_{T}/A_{T}} \right)^{\frac{3}{4}}},{\left( L_{MS}^{3} \right)^{\frac{2}{3}} \propto \left( \left( {I_{MS}/A_{MS}} \right)^{\frac{3}{4}} \right)^{\frac{2}{3}}},{L_{MS}^{2} \propto \left( {I_{MS}/A_{MS}} \right)^{\frac{1}{2}}},{L_{T}^{2} \propto \left( {I_{T}/A_{T}} \right)^{\frac{1}{2}}}$Assuming that z and ρ are constant:

$\frac{\sigma_{MS}}{\sigma_{T}} \propto \frac{A_{MS}{I_{T}\left( {I_{MS}/A_{MS}} \right)}^{\frac{1}{2}}}{A_{T}{I_{MS}\left( {I_{T}/A_{T}} \right)}^{\frac{1}{2}}} \propto \frac{\sqrt{A_{MS}I_{T}I_{MS}}}{\sqrt{A_{T}I_{MS}I_{T}}} \propto \sqrt{\frac{A_{MS}}{A_{T}}}$This indicates that for the same oscillation frequency, the reduced massfiber has a longer cantilever length.

FIG. 6 provides a plot of the merit function showing gain in pointingangle for a microstructured optical fiber over a solid or conventionaloptical fiber based on a fiber outside diameter of 125 μm based on theabove equations. Although this example describes mass reduction in termsof air-filled regions, it will be appreciated that other mass reductionmaterials may be used under a similar analysis. The plot provides asurface showing maximum gain in pointing angle of an optical fiber as afunction of the ratio of diameter to pitch of air-filled regions and asa function of the diameter of air-filled regions, and indicates theeffect of the fiber. It will be appreciated that the maximum gainpossible is desirable, but certain mechanical considerations must betaken into account. For example, the fiber must be able to oscillatewithout breaking. As such, the diameter of the microstructured areashould be less than the outside diameter of the fiber in order for thefiber to have mechanical integrity. Additionally, the ratio of thediameter of the air-filled regions to the pitch of the air-filledregions should be less than 1, otherwise the air filled regions wouldoverlap and occupy an unusable amount of the fiber. As illustrated inFIG. 6, a diameter of the microstructured (air-filled) region of 102.9μm, with a diameter/pitch of 0.873 is anticipated to provide a gain of1.725, which corresponds to a 72.5% increase in pointing angle.

Example 2: Microstructured Fiber Performance Analysis

The requirement for maximizing the natural frequency of a mechanicalsystem is straightforward. For a lumped parameter system, such as a massconnected to a single degree of freedom spring free from damping, thenatural frequency, f_(n) (Hz), is a function of the modal stiffness,

${k\left( \frac{N}{m} \right)},$and modal mass, m (kg). The natural frequency is

$\begin{matrix}{f_{n} = {\frac{1}{2\;\pi}\sqrt{\frac{k}{m}}}} & (1)\end{matrix}$In this case, for given constraints on the problem, there are only twoparameters and either the stiffness can be increased or the massdecreased to boost the natural frequency. For a real, continuous system,like the cantilevered beam the fiber scanner illustrated in FIG. 1B(having a modulus E, density ρ, cross-sectional area A, and secondmoment of area I), the mass distribution, material properties, boundaryconditions (holding method for the fiber) and the relationship betweenbeam dimensions, and the operational parameters of the display (field ofview, refresh rate, resolution) also should be considered. It is alsopossible to change the geometry function of the length of the beam. Thisis done classically in the case of a tapered beam, but is not consideredhere. Rather, microstructured or photonic bandgap type fibers areexamined for their potential benefits. A brief background of oscillatorsis first provided.

The natural frequency equation for lateral motion of an Euler BernoulliBeam is

$\begin{matrix}{f_{n} = {{\frac{\beta\;{L(i)}^{2}}{2\;\pi}\sqrt{\frac{EI}{\rho\;{AL}^{4}}}} = {{\frac{\beta\;{L(i)}^{2}}{2\;\pi}\sqrt{\frac{EI}{\left( {\rho\;{AL}} \right)L^{3}}}} = {{\frac{\beta\;{L(i)}^{2}}{2\;\pi\; L^{2}}\sqrt{\left( \frac{E}{\rho} \right)\left( \frac{I}{A} \right)}} = {\frac{\beta\;{L(i)}^{2}}{2\;\pi\; L^{2}}R_{g}\sqrt{SM}}}}}} & (2)\end{matrix}$where:α: Pointing angle and end of cantilevered fiberI, I_(T,MS): 2nd moment of area of cross sectionT: Subscript for standard, straight, cylindrical fiberMS: Subscript for micro structured fiberA, A_(T,MS): Area of cross section of cantilever (constant overlength—i.e. no tapered fibers)A_(H_i): Area of hole in micro structure (not constant in some designs)r_(i): Perpendicular distance from hole to neutral axisT: Subscript to indicate solid, traditional fiberMS: Subscript to indicate micro structured fiberi: Vibrational mode number (e.g. 1st, 2nd, 3rd mode of vibration andnatural frequency)β(i): Mode constant for i^(th) mode, depends on boundary conditions(cantilevered, free, simple support)D: Oscillator (fiber) outside diameterd: Oscillator (fiber) inside diameter for a tubeδ: Lateral beam deflectionL: Length of cantileverE: Young's modulusρ: Densityfn: Natural frequency of beam—only consider 1^(st) modeSM: Specific Modulus (or specific stiffness),

$\frac{E}{\rho}$R_(g): Radius of gyration,

$\sqrt{\frac{I}{A}}$ξ: Damping ratio of oscillatorThe radius of gyration is defined above as

$\begin{matrix}{R_{g} = \sqrt{\frac{I}{A}}} & (3)\end{matrix}$By inspecting Equation 2, the effects of the independent variables onthe beam natural frequency can be recognized as follow:

Increasing Young's Modulus (E), increases natural frequency

Increasing second moment of area (I), increases natural frequency

Increasing the density, decreases the natural frequency

Increasing the area of the cross section (A), decreases the naturalfrequency

Increasing the length (L), decreases the natural frequency

The natural frequency varies linearly with the mode constant (β). Thisconstant is a function of the boundary condition.

Any one or more of the following may be performed to boost the naturalfrequency.

Modify Boundary Conditions.

Change the wave guide holding methods to increase the constant βL(i).Holding methods are limited by stability requirements, drive energycoupling, and fabrication techniques.

Modify Material Properties.

Increase Young's modulus (E). This is difficult, as optical wave guidematerials are limited.

Reduce the density (ρ). This is also difficult, as optical wave guidematerials are limited particularly in the context of mass production.

These objectives are combined by introducing the idea of specificmodulus or specific stiffness defined as

$\left( \frac{E}{\rho} \right).$Thus, it may be desirable to maximize the specific stiffness, thoughthis is limited by commercial materials. This balances change inmaterial properties to allow direct material comparison.

Modify Mass Distribution.

Reduce the cantilever length (L). Although this is possible, such achange may reduce lateral deflection (static deflection for an appliedend load is proportional to L³.

Reduce the area of the cross sectional (A). This reduces mass, but alsoreduces second moment of area (I) and, correspondingly, stiffness.

Increase the second moment of area (I). This increases the naturalfrequency to the extent that the increase in second moment of area isgreater that the increase in area. Boosts the cross sectional area andthus the mass which reduces the natural frequency. Thus, the ratio of Ito A is an important factor here

$\left( \frac{I}{A} \right).$Thus, it is advantageous to normalize the second moment of area (I)relative to the (A). Recall from Equation 3 that this is the definitionof the radius of gyration. This is a useful approach in designingmicro-structured oscillators. See Equation 14.So, as with material properties, it is useful to normalize the massdistribution effect on natural frequency. Advantageously, this is doneby the property of radius of gyration which is used to describe thedistribution of area about a central axis.

From Equation 2, the proportionality relationships that define theoscillator design trade space (elements dropped from the relationshipare constant) may be used. The proportionally relations are used sincethe relative performance between different oscillator types is ofinterest. Based on Equation 2:

In terms of the natural frequency of the beam, f_(n),

$\begin{matrix}{f_{n} \propto \frac{R_{g}\sqrt{SM}}{L^{2}}} & (4) \\{f_{n} \propto R_{g}} & (5) \\{f_{n} \propto \sqrt{SM}} & (6) \\{f_{n} \propto \frac{1}{L^{2}}} & (7)\end{matrix}$For a fixed frequency, the cantilever length (L) is

$\begin{matrix}{L \propto \sqrt{R_{g}}} & (8) \\{L^{2} \propto \frac{1}{f_{n}}} & (9) \\{L \propto \frac{1}{\sqrt{fn}}} & (10)\end{matrix}$

The frequency gain—between different oscillators—is proportional to thesquare root of the specific modulus, the radius of gyration (square rootof I/A), the boundary constant

$\frac{\beta\;{L(i)}^{2}}{2\pi\; L^{2}}$and inversely proportional to the square root of the cantilever length.

In the fiber scanner application, it is desirable to simultaneouslymaximize natural frequency (f_(n)) and lateral deflection (δ). Themaximum achievable deflection is identified in Equation 11, where, ingeneral, an increase in natural frequency corresponds to a reduction inlateral deflection. Thus, changes in the length are not always useful inincreasing overall system performance without other changes.

Recall that for a quasi-static analysis, the deflection of acantilevered beam with a distributed load per unit length (w) is

$\begin{matrix}{\delta = {\frac{{wL}^{4}}{8{EI}} = {\frac{\rho \cdot A \cdot L^{4}}{8 \cdot E \cdot I} = \frac{L^{4}}{8 \cdot {SM} \cdot R_{g}^{2}}}}} & (11)\end{matrix}$However, the distributed load is proportional to the cantilever length,(L¹), thus the deflection (δ) and the cantilever length are related asδ∝L ³  (12)Thus, a merit function can be written in terms of the radius of gyration(R_(g)) from equation 8:

$\begin{matrix}{{\delta \propto \left( R_{g} \right)^{\frac{3}{2}}} = \left( \frac{I}{A} \right)^{\frac{3}{4}}} & (13) \\{{\alpha \propto \left( R_{g} \right)^{\frac{2}{2}}} = {R_{g} = \sqrt{\frac{I}{A}}}} & (14)\end{matrix}$For comparing two oscillators (subscripts 1, 2) with the same naturalfrequency, the deflection gain is:

$\begin{matrix}{\frac{\delta_{2}}{\delta_{1}} \propto \left( \frac{R_{g_{2}}}{R_{g_{1}}} \right)^{\frac{3}{2}}} & (15)\end{matrix}$and the pointing angle gain is

$\begin{matrix}{\frac{\alpha_{2}}{\alpha_{1}} \propto \frac{R_{g_{2}}}{R_{g_{1}}}} & (16)\end{matrix}$These merit functions are applied to micro-structured fibers to explorethe relative gain expected compared with traditional, untapered fiberoptic oscillators.

The radius of gyration for a hollow fiber with outside diameter (D) andinside diameter (d) is:

$\begin{matrix}{R_{{g\;}_{\_\;{MS}}} = \frac{\sqrt{D^{2} + d^{2}}}{4}} & (17)\end{matrix}$Thus, for micro-structured fiber, the highest value of the radius ofgyration is achieved for a thin walled tube.

The radius of gyration for a solid fiber is:

$\begin{matrix}{R_{g_{\_\; T}} = \frac{D}{4}} & (18)\end{matrix}$Since the natural frequency of an oscillator is proportional to theradius of gyration, this parameter can be maximized for a specifiedfiber diameter.The second moment of area is

$\begin{matrix}{I = {\frac{\pi}{64}\left( {D^{4} - d^{4}} \right)}} & (19)\end{matrix}$The cross-sectional fiber area is

$\begin{matrix}{A = {\frac{\pi}{4}\left( {D^{2} - d^{2}} \right)}} & (20)\end{matrix}$

To illustrate the gain achieved by microstructured fibers, examples ofmicrostructured sections (limits) based on some common fiber sizes andthin wall tubes are considered. A solid fiber of 125 μm diameter and asimplified model of a microstructured fiber of a 125 μm diameter with a10 μm wall thickness (inside diameter 105 μm) is considered.

The second moment of area for the solid fiber is

$\begin{matrix}{I_{T} = {{\frac{\pi}{64}\left( {125^{4} - 0^{4}} \right)} = {{12 \cdot 10^{6}}\mspace{11mu}{µm}^{4}}}} & (21)\end{matrix}$The cross-sectional area for the solid fiber is

$\begin{matrix}{A_{T} = {{\frac{\pi}{4}\left( {125^{2} - 0^{2}} \right)} = {12,300\mspace{11mu}{µm}^{2}}}} & (22)\end{matrix}$The second moment of area of the micro-structured fiber is

$\begin{matrix}{I_{MS} = {{\frac{\pi}{64}\left( {125^{4} - 105^{4}} \right)} = {{6 \cdot 10^{6}}\mspace{11mu}{µm}^{4}}}} & (23)\end{matrix}$The cross-sectional area for the micro-structured fiber is

$\begin{matrix}{A_{MS} = {{\frac{\pi}{4}\left( {125^{2} - 105^{2}} \right)} = {3,610\mspace{14mu}{µm}^{2}}}} & (24)\end{matrix}$The radius of gyration ratio is

$\begin{matrix}{\frac{R_{g\;\_\;{MS}}}{R_{g\;\_\; T}} = {\sqrt{\frac{I_{MS}}{A_{MS}}\frac{A_{T}}{I_{T}}} = {\sqrt{\frac{6 \cdot 10^{6}}{3,610}\frac{12,300}{12 \cdot 10^{6}}} = 1.31}}} & (25)\end{matrix}$

Using the relative deflection gain from Equation 14, and substitutingthe result from Equation 23 to get the best-case gain for a 125 μmoutside diameter fiber (hollow) with a 10 μm wall thickness, thedeflection gain for the micro-structured fiber compared to the solidfiber is

$\begin{matrix}{{\frac{\delta_{MS}}{\delta_{T}} \propto \left( \frac{R_{g_{MS}}}{R_{g_{T}}} \right)^{\frac{3}{2}}} = {(1.31)^{\frac{3}{2}} = 1.5}} & (26)\end{matrix}$Accordingly, this change results in a ˜50% increase in deflection for agiven natural frequency and material.

Now, all parameters for the two designs are determined to verify.Solving Equation 2 for the cantilever length:

$\begin{matrix}{L = \sqrt{\frac{\beta\;{L(i)}^{2}}{2{\pi \cdot f_{n}}}\sqrt{\frac{EI}{\rho\; A}}}} & (27)\end{matrix}$Calculating the cantilever length for the solid fiber:

$\begin{matrix}{{L_{T} = {\sqrt{\left( {\frac{(1.875104)^{2}}{2{\pi \cdot 25}\text{,}000}\sqrt{\frac{{73 \cdot 10^{9}}{{Pa} \cdot 12 \cdot 10^{6}}\mspace{11mu}{{µm}^{4} \cdot \left( {{1 \cdot 10^{- 6}}\frac{m^{2}}{{µm}^{2}}} \right)^{2}}}{2200{\frac{Kg}{m^{3}} \cdot 12}\text{,}{300 \cdot {µm}^{2}}}}} \right)} = {2.007\mspace{14mu}{mm}}}}\quad} & (28)\end{matrix}$Calculating the cantilever length for the micro-structured fiber:

$\begin{matrix}{{L_{MS} = {\sqrt{\left( {\frac{(1.875104)^{2}}{2{\pi \cdot 25}\text{,}000}\sqrt{\frac{{73 \cdot 10^{9}}\mspace{11mu}{{Pa} \cdot 12 \cdot 10^{6}}{{µm}^{4} \cdot \left( {{1 \cdot 10^{- 6}}\frac{m^{2}}{{µm}^{2}}} \right)^{2}}}{2200{\frac{Kg}{m^{3}} \cdot 3}\text{,}{610 \cdot {µm}^{2}}}}} \right)} = {2.293\mspace{14mu}{mm}}}}\quad} & (29)\end{matrix}$Verifying the merit function, these results are compared using Equation8:

$\begin{matrix}{{\frac{L_{T}}{L_{MS}} \propto \frac{\sqrt{R_{g_{T}}}}{\sqrt{R_{g_{MS}}}}} = {\sqrt{1.31} = {1.15 = \frac{2.293\mspace{14mu}{mm}}{2.007\mspace{14mu}{mm}}}}} & (30)\end{matrix}$

Finally, to check the results, the expected deflection is compared for aconstant load using the quasi static analysis as a merit function fromEquation 24:

$\begin{matrix}{\frac{\delta_{MS}}{\delta_{T}} = {\frac{L_{MS}^{3}}{L_{T}^{3}} = {\frac{\left( {2.293\mspace{14mu}{mm}} \right)^{3}}{\left( {2.007\mspace{14mu}{mm}} \right)^{3}} = 1.499}}} & (29)\end{matrix}$Finally, applying Equation 14 directly based on the radius of gyrationration from Equation 23 provides the deflection gain:

${\frac{\delta_{2}}{\delta_{1}} \propto \left( \frac{R_{g_{2}}}{R_{g_{1}}} \right)^{\frac{3}{2}}} = {(1.31)^{\frac{3}{2}} = 1.499}$

Note that the ˜1.5× gain described here is based on an ideal 10 μm wallthickness for a fiber with an outside diameter of 125 μm. In practice,the results for a real fiber with the same dimensions are somewhat less.

Equation 2 may be solved for the length as

$\begin{matrix}{{L^{2} = {\frac{\beta\;{L(i)}^{2}}{2\pi\; f_{n}}R_{g}\sqrt{SM}\mspace{14mu}{or}}}{L = {\sqrt{\frac{\beta\;{L(i)}^{2}}{2\pi\; f_{n}}R_{g}\sqrt{SM}} = {\beta\;{L(i)}\sqrt{\frac{{R_{g}({SM})}^{0.5}}{2\pi\; f_{n}}}}}}} & (30)\end{matrix}$From this

$\begin{matrix}{{{L \propto \sqrt{R_{g}}} = {\sqrt[4]{\frac{I}{A}}\mspace{14mu}{and}}}{{L \propto \sqrt[4]{SM}} = \sqrt[4]{\frac{E}{\rho}}}} & {(31)\;}\end{matrix}$From Equation 31, it is evident that the length is maximized when theratio of the second moment of area to the area is maximized.

As another example, an solid optical fiber with an outer diameter of 80μm and a natural frequency of about 60 kHz is considered and comparedwith a microstructured optical fiber having an overall air fill fractionof about 43.7% (modeled, as above, with a 80 μm fiber with a 10 μm wallthickness) and a natural frequency of about 60 kHz to determine theincrease in pointing angle.

The second moment of area for the microstructured fiber is

$\begin{matrix}{I_{TMS} = {{\frac{\pi}{64}\left( {80^{4} - 60^{4}} \right)} = {{1.374 \cdot 10^{6}}\mspace{14mu}{µm}^{4}}}} & (32)\end{matrix}$The cross-sectional area for the microstructured fiber is

$\begin{matrix}{A_{TMS} = {{\frac{\pi}{4}\left( {80^{2} - 60^{2}} \right)} = {2200\mspace{14mu}{µm}^{2}}}} & (33)\end{matrix}$The length for the microstructured fiber is:

$\begin{matrix}{{L_{MS} = {\sqrt{\left( {\frac{(1.875104)^{2}}{2{\pi \cdot 25}\text{,}000}\sqrt{\frac{\begin{matrix}{{73 \cdot 10^{9}}\mspace{11mu}{{Pa} \cdot 1.374 \cdot 10^{6}}\mspace{11mu}{{µm}^{4} \cdot}} \\\left( {{1 \cdot 10^{- 6}}\frac{m^{2}}{{µm}^{2}}} \right)^{2}\end{matrix}}{2200{\frac{Kg}{m^{3}} \cdot 2200 \cdot {µm}^{2}}}}} \right)} = {1.159\mspace{14mu}{mm}}}}\quad} & (34)\end{matrix}$The second moment of area for the solid fiber is

$\begin{matrix}{I_{T} = {{\frac{\pi}{64}\left( {80^{4} - 0^{4}} \right)} = {{2.01 \cdot 10^{6}}\mspace{14mu}{µm}^{4}}}} & (35)\end{matrix}$The cross-sectional area for the solid fiber is

$\begin{matrix}{A_{T} = {{\frac{\pi}{4}\left( {80^{2} - 0^{2}} \right)} = {5025\mspace{14mu}{µm}^{2}}}} & (36)\end{matrix}$The length for the solid fiber is

$\begin{matrix}{{L_{T} = {\sqrt{\left( {\frac{(1.875104)^{2}}{2{\pi \cdot 60}\text{,}000}\sqrt{\frac{{73 \cdot 10^{9}}\mspace{11mu}{{Pa} \cdot 2.01 \cdot 10^{6}}\mspace{11mu}{{µm}^{4} \cdot \left( {{1 \cdot 10^{- 6}}\frac{m^{2}}{{µm}^{2}}} \right)^{2}}}{2200{\frac{Kg}{m^{3}} \cdot 5025 \cdot {µm}^{2}}}}} \right)} = {1.037\mspace{14mu}{mm}}}}\quad} & (37)\end{matrix}$The ratio of the lengths of the microstructured fiber to the solid fiberLMS/LT is 1.12. Using equation 16, the relative increase in pointingangle is

$\begin{matrix}{{{\frac{\alpha_{MS}}{\alpha_{T}} \propto \frac{R_{g_{MS}}}{R_{g_{T}}}} = {\sqrt{\frac{I_{MS}A_{T}}{I_{T}A_{MS}}}\sqrt{\frac{{1.374 \cdot 10^{6}}\mspace{11mu}{{µm}^{4} \cdot 5025}\mspace{11mu}{µm}^{2}}{{2.01 \cdot 10^{6}}\mspace{11mu}{{µm}^{4} \cdot 2200}\mspace{14mu}{µm}^{2}}}}}\quad} & (38)\end{matrix}$Thus, by microstructuring the 80 μm fiber, the pointing angle can beincreased by 25%.

Example 3: Example Optical Fiber for Scanning Fiber Display

This example describes a microstructured optical fiber embodimentincluding multiple mass reduction regions and use of the optical fiberin a scanning fiber display. The microstructured optical fiber is madeby stacking lengths of optical silica tubes to form an overall preformstructure. A single solid tube of optical material is positioned at thecenter of the preform and used to correspond to the core of awaveguiding region of the final formed optical fiber. A series of solidsilica tubes are positioned around the solid tube in the preform andused to correspond to the cladding layer of the waveguiding region ofthe final formed optical fiber. Hollow silica tubes are positionedaround the series of solid silica tubes in the preform and used tocorrespond to mass adjustment regions of a mechanical region of thefinal formed optical fiber. Finally a ring of solid silica tubes arepositioned on an outside of the hollow silica tubes and used tocorrespond to an outer solid edge or periphery of the mechanical regionof the final formed optical fiber. The assembled preform is heated tofuse the components to one another and then is drawn into an opticalfiber according to known optical fiber drawing techniques.

The resultant optical fiber corresponds to a microstructured opticalfiber. The resultant optical fiber has a core region, such as havingabout a 5 μm diameter core, a cladding region, such as having about a 25μm outer diameter, and a mechanical region having about an 80 μmdiameter. The microstructured optical fiber exhibits an overallair-filled fraction, for example, of about 44%. The microstructuredoptical fiber exhibits a corresponding reduced mass in the mechanicalregion as compared to an equivalent non-microstructured optical fiber.

The scanning fiber display is created by positioning the microstructuredoptical fiber in a cantilevered configuration with respect to amechanical actuator, such that a length of the microstructured opticalfiber is free (i.e., unsupported). The length of the unsupported portionof the microstructured optical fiber is about 1.159 mm. The 1.159 mmcantilevered microstructured optical fiber exhibits a resonant frequencyof about 60 kHz. A maximum pointing angle of the microstructured opticalfiber when oscillating at the resonant frequency is about 12.5 degrees.

As a comparison, an equivalent non-microstructured (i.e., solid) opticalfiber in a cantilevered configuration that has an about 60 kHz resonantfrequency (i.e., the same resonant frequency as the above describedmicrostructured optical fiber) is of about 1.037 mm in length and has amaximum pointing angle of about 10 degrees.

What is claimed is:
 1. An optical fiber comprising: a waveguidingelement extending along an axis; a mechanical region surrounding thewaveguiding element, wherein the mechanical region is positioned betweenthe waveguiding element and an outer periphery, and wherein themechanical region comprises a first material having a first density; anda plurality of mass adjustment regions positioned within the mechanicalregion, wherein the plurality of mass adjustment regions comprise asecond material having a second density less than the first density,wherein the plurality of mass adjustment regions are arranged within themechanical region such that the optical fiber exhibits a percentdifference between perpendicular moments of inertia of about 0.4% orless.
 2. The optical fiber of claim 1, wherein the waveguiding elementcomprises a central core region and a cladding layer surrounding thecentral core region.
 3. The optical fiber of claim 2, wherein thecladding layer comprises the first material, and wherein the centralcore region comprises a third material.
 4. The optical fiber of claim 2,wherein the cladding layer and the mechanical region comprise a unitarybody.
 5. The optical fiber of claim 1, wherein the waveguiding elementcomprises a plurality of core regions and a cladding layer surroundingthe plurality of core regions.
 6. The optical fiber of claim 1, whereinthe plurality of mass adjustment regions comprises one or moregas-filled regions, air-filled regions, one or more polymer-filledregions, one or more glass-filled regions, one or more evacuatedregions, or any combination of these.
 7. The optical fiber of claim 1,wherein the plurality of mass adjustment regions comprises a pluralityof rows of mass adjustment elements, wherein the plurality of rows ofmass adjustment elements are arranged concentrically around thewaveguiding element.
 8. The optical fiber of claim 1, wherein theplurality of mass adjustment regions are arranged in a symmetricconfiguration around the axis.
 9. The optical fiber of claim 1, whereinthe plurality of mass adjustment regions are arranged with axes parallelto the axis.
 10. The optical fiber of claim 1, wherein the plurality ofmass adjustment regions occupy between about 30% and about 90% of avolume of the mechanical region.
 11. The optical fiber of claim 1,wherein the plurality of mass adjustment regions reduces a mass of theoptical fiber per unit length as compared to a comparable optical fibercomprising a corresponding waveguiding element that is identical to thewaveguiding element and a corresponding mechanical region that isidentical to the mechanical region except that the correspondingmechanical region does not include mass adjustment regions positionedbetween the corresponding waveguiding element and a corresponding outerperiphery of the comparable optical fiber.
 12. The optical fiber ofclaim 1, wherein the optical fiber has an effective cantilever length,and wherein the plurality of mass adjustment regions increases aresonant oscillatory frequency of the optical fiber as compared to acomparable optical fiber having the effective cantilever length andcomprising a corresponding waveguiding element that is identical to thewaveguiding element and a corresponding mechanical region that isidentical to the mechanical region except that the correspondingmechanical region does not include mass adjustment regions positionedbetween the corresponding waveguiding element and a corresponding outerperiphery of the comparable optical fiber.
 13. The optical fiber ofclaim 1, wherein the plurality of mass adjustment regions increases aneffective cantilever length of the optical fiber for a given operatingfrequency as compared to a comparable optical fiber comprising acorresponding waveguiding element that is identical to the waveguidingelement and a corresponding mechanical region that is identical to themechanical region except that the corresponding mechanical region doesnot include mass adjustment regions positioned between the correspondingwaveguiding element and a corresponding outer periphery of thecomparable optical fiber.
 14. The optical fiber of claim 1, wherein theoptical fiber has a resonant frequency, and wherein the plurality ofmass adjustment regions increases an effective cantilever length of theoptical fiber as compared to a comparable optical fiber having theresonant frequency and comprising a corresponding waveguiding elementthat is identical to the waveguiding element and a correspondingmechanical region that is identical to the mechanical region except thatthe corresponding mechanical region does not include mass adjustmentregions positioned between the corresponding waveguiding element and acorresponding outer periphery of the comparable optical fiber.
 15. Ascanning fiber display comprising: an optical fiber, wherein the opticalfiber includes a waveguiding element extending along an axis; amechanical region surrounding the waveguiding element, wherein themechanical region is positioned between the waveguiding element and anouter periphery, and wherein the mechanical region comprises a firstmaterial having a first density; and a plurality of mass adjustmentregions positioned within the mechanical region, wherein the pluralityof mass adjustment regions comprise a second material having a seconddensity less than the first density, wherein the plurality of massadjustment regions are arranged within the mechanical region such thatthe optical fiber exhibits a percent difference between perpendicularmoments of inertia of about 0.4% or less; and an actuator in mechanicalcontact with the optical fiber, the actuator for inducing an oscillationof the optical fiber.
 16. The scanning fiber display of claim 15,wherein the actuator comprises a piezoelectric transducer, anelectromagnetic voice coil, or a thermal actuator.
 17. The scanningfiber display of claim 15, wherein the actuator comprises atwo-dimensional actuator for controlling motion of an end of the opticalfiber in two dimensions.
 18. The scanning fiber display of claim 15,further comprising a visible optical source in optical communicationwith the waveguiding element of the optical fiber.
 19. The scanningfiber display of claim 15, further comprising a multi-color switchableoptical source in optical communication with the waveguiding element ofthe optical fiber.
 20. An optical fiber comprising: a waveguidingelement extending along an axis; a mechanical region surrounding thewaveguiding element, wherein the mechanical region is positioned betweenthe waveguiding element and an outer periphery, and wherein themechanical region comprises a first material; and a plurality of secondmoment of area adjustment regions positioned within the mechanicalregion, wherein the plurality of second moment of area adjustmentregions are arranged within the mechanical region such that the opticalfiber exhibits a percent difference between perpendicular moments ofinertia of about 0.4% or less, and wherein the plurality of secondmoment of area adjustment regions comprise a second material forgenerating a second moment of area of the mechanical region that isdifferent from that of a comparable optical fiber comprising acorresponding mechanical region that is identical to the mechanicalregion except that the corresponding mechanical region does not includethe plurality of second moment of area adjustment regions positionedbetween a corresponding waveguiding element and a corresponding outerperiphery of the comparable optical fiber.